THE COMPARISON OF LIFE TABLE AND MARKOV CHAIN
TECHNIQUES FOR FOLLOW-UP STUDIES
by
John R.
Schoenfelder, Richard H. Shachtman
and Gordon J. Johnston
Department of Bio~ta~istic$
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 1299
July 1980
THE COMPARISON OF LIFE TABLE AND MARKOV CHAIN
TECHNIQUES FOR FOLLOW-UP STUDIES
by
John R. Schoenfelder, Richard H. Shachtman and
Gordon J.
Johnston
ABSTRACT
We examine Markov chain versus life table
techni~ues
for comparing
cohorts (homogeneous groups) with respect to time to occurrence of
specified events.
Key results are (1) a theorem showing that, when
the underlying process is a stationary Markov chain (MC),
the
MC
esti-
mator of time to occurrence is equivalent to the life table estimator
and (2) simulation statistics suggesting that the
smallar standard error.
estimator has a
We supplement these features with a discussion of
further advantages associated with using a
Key Words:
MC
Me
modeling approach.
Markov Chain, Life Table, Fallow-up Studies,
Time to Absorption (Occurrence) Distributions
Richard H. Shachtman is Professor, Department of Biostatistics,
•
Curriculum in Operations Research and Systems Analysis, University of
,
North Carolina, Chapel
John R.
H~ll,
NC 27514.
Schoenfelder is Visiting Research Assistant Professor,
Department of Biostatistics, University of North Carolina, Chapel Hill,
NC 27514.
Gordon J.
Johnston is Systems Engineer, Texas Instruments,
Lewisville, TX 75067
Research was supported in part by National Institute of Child
Health and Human Development Grant HD 10367.
1
THE COMPARISON OF LIFE TABLe AND MARKOV CHAIN
TECHNIGUES FOR FOLLOW-UP STUDIES
•
by
John R. Schoenfelder, Richard H. Shachtman and
Gordon J.
1.
Johnston
INTRODUCTION
Biological and epidemiological cohort studies often investigate
the length of time between the occurrences of two specified events.
Usually these events are the initiation of observation and the
developm~nt
of a chronic condition whose incidence is being scrutinized.
The probability distribution of the resulting interval length represents
the time-to-appearance of the condition.
•
For example, for a study of
the incidence of lung cancer in adults, observations might initiate
at the 21st birthday of each subJect and the condition attribute recorded
would be the date of the first diagnosis of lung cancer.
The resulting
distribution of time-to-appearance of lung cancer will then reflect the
propensity of members of the population represented by the study cohort
to contract the disease.
A traditional approach to estimating this probability distribution
...
is to construct a current life table, Chiang (1968), with the appearance of the condition, e. g.,
•
lung cancer, representing the death state.
Such a life table (LT) represents the experience, with respect to development of the condition, of the study cohort during the period of
observatio~
and uses that representation to project the experience of the target
population.
2
An alternative methodology for estimating the time to appearance
probability distribution is proposed by Shachtman, Schoenfelder and Hogue
(1978) in a study of 'the effect of induced abortion on sUbsequent
pregnancy outcome.
Their approach requires that the stochastic nature
of the process being investigated be described by a stationary <time
homogeneous) Markov Chain <MC).
et al.
l~hen
this is the case, Shachtman
(1978) show how functions of the
MC
transition probability
0'
matrix may be used to compute the probability distribution
I eng th of time between th e ap p earanc e of any two events,
the
i. e.,
length of time between visits to any two states in the
Me
the
state
spac e.
In this paper we formally compare these two methodologies.
In particular, we show that when the underlying process is a stationary
MC,
the
MC
estimator is eqUivalent to the
LT
estimator in
the sense that both quantities estimate the same theoretical construct.
The
MC
estimator,
however, makes use of the Markov property whereas
the
LT
estimator does not; hence, as will be evidenced later, the
former makes more efficient use of available data.
In addition, we present several additional advantages which
favor
Me
technology over its
LT
counterpart.
For example; the
MC
approach may be used to compute the probability distribution of the
length
0'
however,
time between visits to any two states of the
MC.
A LT,
produces only the distribution function of the length of time
between the initial state and (the state representing) the appearance
0'
the condition.
Thus one
MC
may be used to investigate several
research questions, whereas each individual research question reqUires
a separate
LT.
Hogue (1980),
the
Furthermore, as seen in Shachtman, Schoenfelder and
MC
formulation provides a straightforward
methodolog~
3
for incorporating intervening variables into the analysis;
is found in the
LT
no analogue
approach.
Before discussing comparisons between
MC
and
LT
we present the mathematical basis for both techniques.
methodologies.
In this
development we shall. for convenience and notational ease. assume that
the condition being studied.
once contracted. may not go into remission -
thus it is represented by an absorbing state in the Markov chain.
Since
the purpose of this work is to estimate the probability distribution of
time until appearance of the condition.
this assumption does not prevent
the methodologies from being used in a study of conditions which may go
In such situations the condition state in the
into remission.
artificially made absorbing for the purpose of analysis.
MC
is
The resulting
probability distributions are called time to absorption (TTA) distribut ions.
•
TIME TO ABSORPTION DISTRIBUTIONS
2.
We begin by developing a mathematical representation of the theoretieal
TTA
distribution which we desire to study.
construct two specific estimators of the
estimator and the
Me
estimator.
TTA
We then define and
distribution - the
LT
In the next section we formally
compare these estimators.
We consider a discrete-time stochastic process
(X : t = 0.
1.
2, ... )
which, at any given time. may occupy one of
t
fin i tel y ma n y s tat e s
1.
2..... K.
Assume that
r
of these states
are absorbing and with no loss of generality further assume that
r
=1
and that state
example,
cane er.)
K
is the absorbing state.
entrance into state
K
In ord er to ob ta in both
(In our introductory
would represent contracting lung
LT
and
MC
es t imators, we assume
4
that the data available for our analysis is a set of statistically
independent sample paths generated by this process.
Let the set of probabilities
(a (0)
J
=
1,2, ...
I
Ki
J
K
a
SUt1
=
(0)
J=l
where
1),
SUM
represents the ordinary arithmetic sum,
J
constitute the initial distribution over these states,
a
=
(0)
P(X
J
o
=
Finally,
J>'
let
i.
e.,
be the random variable representing
T
the time (measured from 0) required for the process to first enter the
absorbing state.
Then the time to absorption probability distribution
which we desire to estimate is the cumulative distribution function of
T,
F
(n)
= P(T <=
To obtain an expression for
n).
T
we define
T
= P(being
q
...
(n)
F
absorbed at time
m + 1
wasn't absorbed by time
m
= P(X
m+1
where
1=
=K
X 1= K),
m
means unequal.
Hence
p
m
= P(not
time
=1
-
m+ 1
being absorbed at time
m)
wasn't absorbed by
q
ITl
and
P(T) n+1)
=1
- F (n+1)
T
n
= PROD
m=O
where
PROD
P
m
=
n
PROD
m=O
(1
-
q ),
m
represents the ordinary arithmetic product.
m)
Thus,
=1
F (n+1)
n
- PROD
(1 - It
m~O
T
(2. 1)
m
....
is an estimator of
F (n + 1), where
T
We now need to construct
It
m
is an estimator of q .
m
Define
It
m
a
=
(m)
P(X
m
J
= J)
P(X
x
=:
m+l
=:
J
>.
m
We may then write
It
m
..
e
=
P(X
=
K-1
SUM
x
=: K
m+l
/= K)
m
P(X
m+l
J=1
1(-1
X = J) / SUM
m
h=1
= K,
=
K-1
SUM
J=l
P(X
=
K-1
SUM
J=1
a (m) r (m)
m+1
/
JK
K-1
SUM
h=:1
= h)
m
X = J) P(X =
m
m
= K
J
P(X
J)
/
K-l
SUM
h=l
a (m)
h
Recalling that the data available to estimate
P(X
m
= h)
(2.2)
are statistically
It
m
independent sample paths, we compute the following quantities:
N
(m)
=
number of individuals in state
J
at time
m
=:
number of individuals in state
in state h at time m + 1.
J
at time
m and
J
N
(mj
Jh
Now,
if
N
is the total number of sample paths then,
for each
m,
K
we
have
SUM
J=l
N (m) = Ni
J
i.
e.
I
we
assume that there are no withdrawals
6
from the study.
of the
TTA
At this point, we develop the
by computing estimators of
LT
and
based on
~
MC
LT
estimators
and
Me
m
methodology.
2.1
The Life Table Estimator
This estimator results from making no assumptions about
the probabilistic structure of
X.
In particular, we define the
estimators
a
(m)
==
N
J
(m)
N
/
J
.....
r
(m)
=N
Jh
(m)
I
N
Jh
(mL
J
From these we use expression (2.2) to obtain
K-l
q, = SUM
m
J=l
K-l
SUt1
K-l
a (m) r
(m) I SUM
J
JK
h=l
A
A
( N ( m) IN)
J=l
J
a
(m)
h
( N ( m) I N ( m) )
JK
J
= -------------------------------K-l
SUM
h=l
N (m) IN
h
K-l
= SUM
J=l
:'<'-1
N
(m)
SUM
I
JK
h=1
N
(2.3)
(m)
h
K-l
In li-Pe table terminology, Chiang (1968),
d
m
is the number "dying"
(i.
e.
I
= SUM
J=l
N
(m)
JK
being absorbed, contracting the condition)
7
Un,
in the interval
K-1
m+l) and
N
m
at time
N (m)
J
J=1
the estimator oT
Thus,
m.
= SUM
F
is the number "alive"
obtained by substituting the
T
.
estimators of
given in (2.3) into expression (2.1) is the
q
m
complement of the conventional
=1
F (n+ 1 )
T
2.2
LT
survivorship estimator; namely
n
- PROD
m=O
(1 -
diN
m
>.
m
(2.4)
The Markov Chain Estimator
Whereas we imposed no assumption on the probabilistic structure
of
X
when deriving the
~
X
is a
That is, we assume that the movement oT the
stationary Markov chain.
..
estimator, we now assume that
LT
process among the states is governed by the transition probability
matrix
P
= «p
»
where
iJ
=
p
=J
P(X
iJ
=i
x
n-1
11
>.
Note that since the underlying process
probabilities do not depend on
In order to obtain the
Me
the following
X
is stationary these
n.
distribution, we need consider
MC TTA
(1) the
probabilities:
n-step
transition
probabilities given by
p
..
(n)
=
m+n
iJ
= i)
x
=J
P(X
m
and (ii) the first passage time probabilities given by
f
(n)
= P( X
iJ
= J;
X
/= J'
m
n
1
<=
m
<=
n-1 : X
o
=
i >.
These probabilities are related through the well-known expression
p
(n)
i J
=
n
SUM
m=l
f
(m ) p
i J \
(n-m);
JJ
8
ThuSI we obtain first passage time probabilities iteratively from n-step
transition probabilities; we derive the latter from the transition
probabilities by making use of the fact that
element of the
n-th
is the
(n)
(it J)-th
iJ
power of
P.
Using these ,uantitie, Shachtman et al.
TTA
p
(1978) show that the
Me
distribution is given by
K-l
F (n+l)
= SUM
a
(2. S)
JK
J
We substitute estimates of
(n+l>'
p
(0)
J=1
T
(n+1) into the above e,uations to
p
JK
obtain the
MC TTA
estimates.
The
(n+1)
p
estimates in turn derive
jK
from the maximum likelihood estimators of the
p
IS
by employing
iJ
the previously presented relationship.
likelihood estimator of
Finally, the maximum
is given by
p
iJ
=
IN .
N
i J
i
where
N
= SUM
m
iJ
N
i.
= SUM
h
N (m)
iJ
N
ih
and
( m)
N
i s asp rev i 0 usly d ef i ned.
iJ
3.
3.1
RELATING THE LIFE TABLE AND MARKOV CHAIN ESTIMATORS
Analytic Comparison
In this section we provide mathematical verification that when the
underlying process
X
is a stationary
Me, the
LT
and
Me
estimators
9
are equivalent in the sense that both estimate the
developed at the beginning of the last section.
that the functional form of the
LT
TTA
distribution
This is done by showing
estimator leads to the
MC
estimator
when the Markov assumption is imposed.
THEOREM:
Assume that the stochastic process governing movement
among a set of states containing an absorbing state is a stationary
Markov chain
Then the life table time to absorption distribution,
derived without benefit of the Markov assumptions,
is equivalent
to the time to absorption distribution computed directly from
the Markov chain.
From expressions (2. 1) and (2.2) the
PROOF:
•
LT TTA
distribution·
is given by
=1 -
F (n+1)
T
n
( i-q
PROD
)
m=O
m
a (m) r
(m) I SUM
where
K-1
:'<'-1
q
m
=
SUM
J==1
Stationarity implies that
r
(m)
a (m).
h
1'1=1
JK
J
does not depend on
m
and,
JK
furthermore,
.
thus
is equivalent to the
MC
transition probability
may be written as
q,
m
K-1
10\-1
= SUM
q
m
J=l
a
(m)
J
p
I
JK
SUM
1'1=1
a
(m).
h
(2.6)
10
Consider the
convention,
n-step
transition probability p
define
p
(0):::
iJ
(n) and, by
iJ
=
1
i
::: J
0
i
1= J.
If we further assume, without loss of generality,
of
being absorbed initially (i. e.,
a
= 0),
(0)
that there is no chance
then
K
1(-1
a
==
(m)
a (0) P
SUM
i=l
J
==
i
(m), and equation (2.6) becomes
i J
K-1
K-1
SUM
SUM
J=1
i=1
(0)
a
P
(m)
p
iJ
i
Denote the denominator by
1
JK
1(-1
1(-1
SUM
SUM a
h=1
k==1
(m ::: 1, 2, ... >.
b
(0 )
k
( m>.
p
kh
Then
m
p
m
=
1(-1
SUM
i==l
K-1
-
1
q
== SUM
m
J=l
(m) ( 1
(0) p
a
i
iJ
-
)
P
JK
1 b
m
K-1
Using
1 - P
==
h=1
JK
:=
P
sur"!
m
and interchanging the summations yields
P
Jh
K-l
K
SUt1
SUM
h=l
i=l
1(-1
a (0 ) SUM
i
J=l
p
== p
P
(m)
iJ
1 b
p
Jh
But,
K
SUM
e
and,
since
(m)
P
J=1
iJ
p
Kh
=0
Jh
for
n
in
(m+1)
1= K,
it follows that
m
11
K-1
P
Thus, using
m
= SUM
K-1
SUM
i=l
=b
/
h=l
p
a (0 ) p (m+1) /
i
ih
m
• 1 - q
F (n+1)
T
=1
=1
-
is expressible as
F
m
= 1-
m
b
m+1
m
b
T
n
PROD
m=O
(l-q )
m
(b
/
b
)
n+1
0
K-1
SUM
i=l
K-1
SUM
h=l
a
(0)
(n+1)
p
i
ih
--------------------------
-
K-1
SUM
J=l
a (0).J
a (0) = 0,
K
Since we have assumed
it follows that
K-1
SUM
J=l
and
(n+1)
F
T
-
.1
= 1
=:
This,
hO'..l1ever,
is the
K-1
K-1
i=!
h-1
K-1
SUM
i=l
a (0) ( 1
i
- SUM SUM
K-1
SUM
i=l
a (0)
=1
J
a (0) p (n+1)
i
ih
-
(n+1) )
p
iK
a (0) p (n+1).
i
iK
Me TTA
distribution as given in equation (2.5>'
###
3.2
Numeric~l
Comparison
Having veri;ied that the theoretical construct being investigated
is the same for both estimation procedures, we now consider the
situation where a researcher has a set of sample paths available
for analysis and must determine which methodology to employ.
If
12
one has reservations about depicting the underlying process as a
stationary
MC
then there is no choice to make;
the
is robust to this assumption since it does not use it.
If, however, one feels confident about the
decision is no longer deterministic.
MC
LT
estimator
/1/
assumption the
Intuitively, since it correctly
incorporates the structure of the process, we feel that the
MC
estimator should be superior.
This intuition is not easily quantifiable.
Because of analytic
difficulties in obtaining standard errors of n-step transition probabilities, there do not exist closed form expressions to construct
confidence intervals for the
MC
estimator; hence we are unable to use
this traditional criterion to compare the estimators.
however,
We have,
performed a simulation study whose results support our
intuition.
In this study we hypothesized several transition matrices and,
with each, associated several initial distributions for a stationary
MC
with five states, one of which was absorbing.
The various
transition matrices were such that some contained relatively uniform
transitiun probabilities and some contained widely varying transition
probabilities.
Likewise the selection of initial distributions encom-
passed vastly differing situations - from uniform over the four transient
states to the situation where all mass was concentrated on one of those
transient states.
let
Retaining the notation of the previous section,
F{n) = P{T <= n)
where
T
is the
time to absorption random
variable being studied.
For each combination of transition matrix and initial distribution
we empluyed the following algorithm:
13
1)
Sim.ulate 100 sample paths, each path of length
2)
Using these sample paths
a.
compute
(n)
(F
n
LT
b.
= 1,2, ... ,50),
the
estimate the transition matrix and,
matrix, compute
(n)
(F
time units.
50
LT
estimator of F.
using that estimated
n == 1, 2, . . . , 50), the
MC
MC
est i ma tor
3)
Repeat steps
4)
Compute (F(n)
0
1
f
F.
and
n =
Thus we had 100 independent
MC
estimates of
for eac h
2
1,
one hundred times.
2, . .. , 50),
LT
the t rue dis t rib uti 0 n .
estimates of
and 100 independent
F
F, based on 10,000 sample paths.
Next we computed,
n,
.....
the'mean
1)
LT
estimate,
F
(n) == (1/100) SUM F
LT
(n)
LT
2
the sample variance of the 100
2)
LT
estimates, S
(n)
LT
3)
the first and ninth deciles of the 100
4)
the mean
MC
es timate,
(n)
F
MC
LT
= (1/100)
estimates.
"
SUM F
(oj
MC
2
5)
the sample variance of the 100
MC
estimates, S
(0 )
MC
6)
the first and ninth deciles of the 100
MC
estimates.
Comparing these statistics we discovered little difference between
F
(n)
and
LT
(n); see Table 1 for the results from one transition
F
MC
matrix and one initial distribution.
Results based on other matrices and
other initial distributions did not vary
the tab Ie.
~ualitatively
Furtheriiiore, both were c lose to
F(n).
from the one in
This suggests little
14
However,
difference between the estimates.
2
2
a IUlay S
I
S
the case that
(n)
<: S
Me
(n) and that the inner decile range
IT
MC
of the
it was usually, although not
estimates Ulas less than the inner decile range of the
estimates; see Figure A.
LT
Since these conclusions held for all transi-
tion matrices and for all initial distributions, these observed results
do not appear to be an artifact of the particular matriX and/or initial
....
distribution simulated.
Thus for each
n, a given
F
(n)
is apparently
t1C
,...
closer to the true
F(n)
than is the corresponding
F
(n).
IT
•
Table 1, Figure A
4.
DISCUSSION
Based on the above results there appears to be little practical
numerical difference in results between these analytic approaches.
Although initially surprising this is understandable as both
procedures are maximum likelihood, although with respect to different
underlying assumptions.
iated with the
Me
There are, however, several advantages as soc-
approach.
The first is that by modeling the entire process, rather than
Just the initial and outcome state, we obtain significantly greater
insight which is useful when interpreting the results.
assume that we compare the
find them to differ.
TTA
If we used
For e xa mp Ie,
distributions for two cohorts and
MC
techniques further investigation
of the transition matrix and/or n-step transition probabilities may
help to explain the difference.
If,
however, we used
IT
techniques,
15
no such additional help is available.
Me
This additional strength of the
approach is available because we model more states oT the process
when obtaining the
TTA
distribution.
When employing
LT
techni~ues,
on the other hand, we omit interim states as consideration of only the
outcome state is suTficient.
(Note that this means we do not need
complete path information to construct a life table and thus the
estimator exerts a lesser data demand than does the
Another advantage is that the
to sensitiVity analysis whereas the
MC
LT
Me
LT
estimator.)
methodology easily lends itself
methodology does not.
Having
estimated the transition matrix, we may postulate changes in (a subset
of) the transition probabilities and compute an altered
tion.
TTA
A comparison oT this distribution with the original
distribuTTA
distri-
bution will assess the extent of the original distribution's dependence
on the altered probabilities.
in
LT
investigations.
There is no apparent analogue to this
Also, any measurements on the
Me
which may
be derived as functions oT the transition probabilities (known as
parameterizations) May be subjected to straightforward sensitiVity
analyses.
In addition since the
Me
allows incorporation of intervening
variables, researchers may evaluate potential intervention strategies.
Using the methodology of Shachtman, Schoenfelder and Hogue (1980),
may employ the
MC
to assess what effect would result on
TTA
one
if one
alters visit patterns to (a subset of the) states, for example, the effect
of an intervention which explicitly prevents the process Trom entering one
aT' more of its states.
Shachtman et a1.
(1976),
1980)
provide an example
of an intervention where intervening variables are contraceptive states in
biological modeling of women's reproductive processes.
analysis is not feasible using a life table.
Again, such an
16
Still another area of superiority of the
Let
N
t ion.
involves prediction.
MC
be the length of the longest observed interval before absorpUsing the
LT
estimator it follows that
F
Cn)
=
LT
all
F
CN)
for
LT
that is, we cannot effectively extrapolate into
n:>= N;
the future.
Such a restriction does not exist for the
MC
estimator
as it is not as severely limited by the lengths of the observed
sample paths;
future.
thus we may use the
MC
to make predictions into the
We must, however, exercise care in interpreting such predictions
as this situation is similar to that of using a fitted regression line
in a domain where there are no observations on the independent variable.
As long as we feel comfortable with the prerequisite assumptions, especially stationarity, this prediction into the future provides a reasonable estimate of the desired distribution.
A final, and possibly most important, benefit associated with
MC
does
analysis is that it offers the researcher more flexibility than
LT
methodology.
Consider the Cnot unusual) situation
w~ere
one
has one data set and desires to address several research questions,
requiring the determination of a
TTA
type distribution function.
After the investigator has estimated the
all of the
TTA
each
Me
transition matrix,
distributions are readily obtainable by computing
functions of the transition probabilities.
investigator employed
LT
Per contra,
if th e
methodology he would need to construct a
separate life table for each research questions.
In summary, we have considered two statistical techniques, one
based on
LT
technology and the other on
MC
techniques, for estimating
the cumulative probability distribution function of the time required
for a stochastic process to reach an absorbing state.
We have shown
that both analytic techniques estimate the same tneoretical distribution;
e
17
furthermore,
resu I ts Or a S1 mu lat i on study i nd i cate tha t th e techn i ques
perform equally well and that there is apparently little practical
numerical difference in results between them.
Yet based on the quali-
tative considerations Just discussed, we feel that if the underlying
assumptions are Justified the
Me
methodology should be employed.
This is not Just because it will produce an estimate which is apparently
closer to "truth ll than would be the
(the
Me
LT
estimate,
but rather because it
methodology) offers the researcher greater flexibility and the
opportunity oT gaining greater insight into the data.
18
FOOTNOTES
/1/
Frequently a redeTinition oT the basic state space Tor the chain
will yield a model which is acceptably close to satisTying the
Markov and stationarity properties.
19
TABLE 1.
RESULTS OF SIMULATION STUDY OF TIME TO ABSORPTION DISTRIBUTION
MONTH
n
MEAN MC
ESTIMATOR
(n)
F
TRUE TTA
DISTRIBUTION
F(n)
Me
5
10
15
20
25
30
35
40
45
50
*
0.246
O. 433
O. 573
0.678
O. 758
0.817
0.862
0.896
O. 921
O. 940
MEAN LT
ESTIMATOR
(n)
F
LT
0.246
0.434
O. 575
0.681
O. 760
0.820
0.865
O. 899
O. 924
0.943
O. 242
0.430
0.573
O. 676
O. 758
0.816
O. 862
0.897
0.923
0.942
based on 100 independent MC and LT est imatesl each oT
which is based on 100 independent sample paths
*
20
FIGURE A.
COMPARISON OF STANDARD ERROR FUNCTIONS FOR
MC
AND
LT
TTA
DISTRIBUTION
ESTIMATORS
.....
:---------J---------~---------i---------:---------:---
o
20
10
30
Months to Absorption
se
Me
se
LT
(n)
=
standard error of
F
(n)
=
standar~
error of
F
Me
(n)
(n)
LT
40
50
-J
21
REFERENCES
Chiang,
Chin Long,
BIOSTATISTICS.
Shachtman,
R. H.
(1968>'
INTRODUCTION TO STOCHASTIC PROCESSES IN
New Vor k,
and Hogue,
John Wi ley and Sons,
C. J.
(1976>.
Inc.
"A Markov Chain Model .por
Events Subsequent to Induced Abortion."
OPERATIONS RESEARCH 24:
916-932.
S hac h t man,
R. H.,
S c hoe n .p e 1 d e r ,
J. R.
and Hog u e ,
C. J.
( 1978 ) .
II
Us i n g a
Stochastic Model to Investigate Time to Absorption Distributions.
II
(To appear in OPERATIONS RESEARCH>
Shachtman, R. H., Schoenfelder,
J. R.
and Hogue,
C. J.
(1980>'
"Condi-
tional Rate Derivation in the Presence of Intervening Variables
Using a Stochastic Model."
No.
1292
Institute of Statistics Mimeo Series
Department OT Biostatistics,
Chapel Hill.
University of North Carolina,
(Submitted to OPERATIONS RESEARCH>